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Theorem mapssfset 8847
Description: The value of the set exponentiation (𝐵m 𝐴) is a subset of the class of functions from 𝐴 to 𝐵. (Contributed by AV, 10-Aug-2024.)
Assertion
Ref Expression
mapssfset (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapssfset
StepHypRef Expression
1 mapfset 8846 . . 3 (𝐵 ∈ V → {𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴))
2 eqimss2 4036 . . 3 ({𝑓𝑓:𝐴𝐵} = (𝐵m 𝐴) → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
31, 2syl 17 . 2 (𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
4 reldmmap 8831 . . . 4 Rel dom ↑m
54ovprc1 7444 . . 3 𝐵 ∈ V → (𝐵m 𝐴) = ∅)
6 0ss 4391 . . 3 ∅ ⊆ {𝑓𝑓:𝐴𝐵}
75, 6eqsstrdi 4031 . 2 𝐵 ∈ V → (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵})
83, 7pm2.61i 182 1 (𝐵m 𝐴) ⊆ {𝑓𝑓:𝐴𝐵}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  {cab 2703  Vcvv 3468  wss 3943  c0 4317  wf 6533  (class class class)co 7405  m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824
This theorem is referenced by: (None)
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